Step-by-step explanation:
[tex]\sin(\theta + \phi) = 2\cos(\theta - \phi)[/tex]
Using the addition identity formulas, we can write
[tex]\sin{\theta}\cos{\phi} + \cos{\theta}\sin{\phi} = 2\cos{\theta}\cos{\phi} + 2\sin{\theta}\sin{\phi}[/tex]
Dividing both sides by [tex]\cos{\phi}[/tex], we get
[tex]\sin{\theta} + \cos{\theta}\tan{\phi} = 2\cos{\theta} + 2\sin{\theta}\tan{\phi}[/tex]
Dividing both sides by [tex]\sin{\theta}[/tex], we get
[tex]\tan{\theta} + \tan{\phi} = 2 + 2\tan{\theta}\tan{\phi}[/tex]
or
[tex]\tan{\theta}(1 - 2\tan{\phi}) = 2 - \tan{\phi}[/tex]
Rearranging the terms,
[tex]\tan{\theta} = \dfrac{2 - \tan{\phi}}{1 - 2\tan{\phi}}[/tex]
